Determine the value of the following complex number power. Your answer will be plotted in orange. $ ({\cos(\frac{23}{12}\pi) + i \sin(\frac{23}{12}\pi)}) ^ {2} $
Solution: Let's express our complex number in Euler form first. $ {\cos(\frac{23}{12}\pi) + i \sin(\frac{23}{12}\pi)} = { e^{23\pi i / 12}} $ Since $(a ^ b) ^ c = a ^ {b \cdot c}$ $ ({ e^{23\pi i / 12}}) ^ {2} = e ^ {2 \cdot (23\pi i / 12)} $ The angle of the result is $2 \cdot \frac{23}{12}\pi$ , which is $\frac{23}{6}\pi$ Our result is $ e^{11\pi i / 6}$. Converting this back from Euler form, we get $\cos(\frac{11}{6}\pi) + i \sin(\frac{11}{6}\pi)$.